# abstract algebra – \$ mathbb {Q} ( zeta_ {10}) \$ field extension over \$ mathbb {Q} \$

I want to calculate the degree of $$mathbb {Q} ( zeta_ {10})$$ over $$mathbb {Q}$$,

Use the dimension set: $$[mathbb{Q}(zeta_{10}) : mathbb{Q}]=[mathbb{Q}(zeta_{10}) : mathbb{Q(zeta_5)}] cdot[mathbb{Q}(zeta_{5}) : mathbb{Q}] = 2 cdot4 = 8$$, Due to the fact that $$x ^ 4 + x ^ 3 + x ^ 2 + x + 1$$ is irreducible and $$x ^ 2- zeta_5$$ is the minimum polynomial for $$zeta_ {10}$$ over $$mathbb {Q} ( zeta_5)$$,

However: $$x ^ 5 + 1 | _ { zeta_ {10}} = {(e ^ { frac {2 pi i} {10}})} ^ 5 + 1 = e ^ { pi i} + 1 = 0$$,

I am confused because the extension should be $$8$$but it seems so $$zeta_ {10}$$ is a root of $$x ^ 5 + 1$$,